Appunti behavioural economics

Choosing When to Act

Choices in which time is important. Time matters because benefits and costs associated to decisions are spread over time. Part of the benefits and costs are not immediate, but in the future.

Suppose that there are T periods (T can be very large); in each period the individual is getting the utility ut. We're not talking about outcomes, but directly about utility. The idea is that an individual wants to transform a sequence of utility over time in a single number which is a measure of intertemporal utility. Utility can be also negative.

Per period utility: utility in which a single period is defined. Rows - decisions taken by Maria (the moment in which she does her homework); the columns - the days. For each day, there is a different utility depending on the action. When Maria does her homework on Friday, the day in which she does her homework, she suffers a cost (-5). On Saturday, Sunday and Monday a positive utility. She does her homework on Saturday, so the

utility on Friday is 0 (no homework done). It's a complete representation of the preferences of Maria, defined for each period of utility associated to each action.

There is a standard way in which preferences are represented. We move from preferences to the intertemporal utility function. The approach is to transform a sequence of utilities into a single number. The intertemporal utility function is a function of the utility in each period.

Exponential discounting = the utility in each period is multiplied by a number (the discount), which is given by the discount factor (δ, between 0 and 1) to the power of the number of periods from today up to the moment in which the utility is obtained. The utility today has a value given just by the utility (there is no discount, it's multiplied by 1). The utility tomorrow is given by u2 (the utility in that period) x delta (δ, between 0 and 1). Delta gives the relative importance that the individual gives to the future (if it is 1,

it means that the utility of tomorrow is equivalent to the utility today, whereas if it's below 1, it means that individual gives it less importance). 10 euro tomorrow is the equivalent to ten x delta euro today. If we consider two periods, we have delta to the power of 2. The same goes for the all the remaining periods. Since delta is smaller than 1, delta's square is smaller than delta -> the higher the exponent, the lower the quantity. The more distant the future, the lower is the importance attached by the individual to that specific period. Delta and rho depend on the definition of what the period is.

The intertemporal utility is different depending on the discount factor. Moving on the right, the individual gives more importance to the present than to the future. You need to apply this formula: to the number of the tab. When δ =1, the optimal choice is to do it on Saturday, where the highest number is (15). If she is not very impatient, she does it on Saturday, while if

The individual is impatient her homework is done later on. Impatient: getting the benefit as soon as possible, while here the benefit means to postpone the action, since doing homework is a cost. Applying discount factors to real world. 4 possible approaches to estimating the discount factors:

• Postpone receipt: if there is some impatience, you want to get more money. How much money depends on the discount factor. If close to one, it would be 200 euro.
• Postpone payment: if the individual has impatience, do you think he is willing to pay less or more after one year? More. You have the possibility to avoid the payment today and pay it tomorrow, so you are willing to pay more tomorrow. How much? It will depend on his discount factor.
• Expedite receipt: a smaller quantity, but how much smaller? If you are impatient, you want to pay less (you are happy to pay tomorrow). This tells us how people tend to think about their intertemporal preferences. On average, the discount factor is

relatively low (0.8, 0.9). This discount factor has an exponent (2, 3…); these numbers get small quite fast—0.9 becomes 0.81 after a period, then 0.27. In the close future, you have quite a low number multiplying utility in that period. The discount factor is higher, the longer it is necessary to wait (short-term impatience). Short-term impatience: the standard approach says that you have a discount factor (0.9) and the importance attached to one period in the future becomes in two period 0.81, then 0.27 and so on. If you consider longer periods to wait, the discount factor you are using is getting higher. Ex. if you consider one period in the future, it is 0.9. If you consider the second period, the importance attached to the second period is not 0.8, but a higher one, like 0.85. If you move in 3 periods, it’s not 0.27, but 0.49 (the discount factor is higher than 0.9). So, the discount factor is not something independent from the period, and this is a violation of exponential

discounting.Exponential discounting is based on the idea that you have 1 parameter (the discount factor) and an exponent, which is the number of periods from the present to the future (the standard approach). So, short-term impatience is the idea that for individuals, there is a big difference between the present and the future. When you consider the future (close or distant), it does not make any difference, whereas the big difference is from the present to the future (the payment from today to in 3 months, but from 3 months in a year there not so much difference).

Absolute magnitude effect: the larger is the sum of money, the larger is the estimated discount factor.

Gain-loss asymmetry: reference points are important to differently treat gains and losses. The estimated discount factor is smaller for gains than for losses.

Delay-speed-up asymmetry: if you ask to postpone or to expedite payments/receipts, and you find that the estimated discount factor is higher to postpone than to expedite payment,

and higher to expedite than to postpone receipts. The discount factor depends on the fact that money is moving from the present to the future and from the future to the present. This should not be the case in a standard rationality approach. The exponential discounting is the benchmark of full rationality. If the trips are close in time, 90% prefers to visit the aunt first and the same is if the trip is in the future. The percentages are almost equal if trips are this weekend and 26 weeks from now (distant in the future). This preference is called preference for an improving sequence. When you have to take two actions close in time, you want to first take the action associated to something negative, and then the beneficial one. This answer is not consistent with exponential discounting if there are 2 actions (one beneficial, one giving negative utility), you want to do first the positive action, because if the negative action is taken in the future is multiplied by a number smaller than one. So,

it’s better to postpone the negative action (it is discounted). It is optimal to have the beneficial action as soon as possible, because it is not discounted.

In a first experimental setting, suppose you are asked to compare only A and B. You can choose between going out to a restaurant this weekend and stay at home the next ones, or staying at home this weekend and go out only the next one. The best thing is to go in the second weekend, since you don’t want the restaurant immediately and have 2 consecutive weeks at home.

Turning to C and D, individuals are sharing the two options quite equally, even if the majority prefers to have the restaurant first and in the last week.

The individual has a preference for spreading the events (the positive things). These results are obtained when the events are closed in time, when the two are distant, then you don’t observe the non-conventional preferences. People think differently when the events are closed, and when they are far away.

Want to have models (stories, explanations) for the deviations that we observe from the standard model. We need to go back to the 5 results of the discount factors (violations of the exponential discounting):

• Short-term impatience - we need to have different ways to define the discount factor, which changes overtime. D (t) is the discount factor in T time periods from now. It's the number which is multiplying the utility in T. With exponential discounting, D (t) is equal to Delta with exponent (t-1). In the graph, delta is constant (blu in the graph) for all the period considered. With hyperbolic discounting (red): D(t) is not constant, but is increasing.

Time consistency: if Maria today prefers 110 euro in 31 days to 100 euro in 30 days (possible if she has shorter impatience), it means that after 30 days, she cannot prefer 100 euro today to 110 euro tomorrow. If you prefer to wait 1 euro today, when the moment comes, you still want to wait. Individuals have present-biased.